A programmer needs to compute inverses of polynomials that have hexadecimal
coefficients other than {00} and {01}. Doctor Vogler helps by clearing up the notation
that appears in the Federal Information Processing Standard (FIPS) Advanced Encryption
Standard (AES).

Is there a way to determine the nature of the roots of a quartic
equation in the form ax^4 + bx^3 + cx^2 + dx = 0 by simply using the
coefficients, as with the discriminant b^2 - 4ac in a quadratic
equation of the form ax^2 + bx + c = 0?

A modern algebra student picks up the thread from another student's earlier
conversation with Doctor Vogler. Together, they re-visit and lay the question to rest,
applying Sturm's Theorem in the process.

A student struggles to conceive of all the binary operations possible in a two-element
set. Doctor Peterson clarifies the scope of the abstraction before enumerating pairs
and offering a template for organizing them.

There are 311 distinct solutions to the equation x^311 = 311x + 311.
These solutions are designated by the 311 variables a_1,a_2,....a_311.
Find the sum (a_1)^311 + (a_2)^311 + (a_3)^311 + ... + (a_311)^311.
I've been told that Newton Sums can be used on this problem, but I'm
not sure how to apply it. Can you help?

Show A_n contains every 3-cycle if n >= 3; show A_n is generated by 3-
cycles for n >= 3; let r and s be fixed elements of {1, 2,..., n} for n
>= 3 and show that A_n is generated by the n 'special' 3-cycles of the
form (r, s, i) for 1 <= i <= n.

Suppose G = {a1, a2, ... , an} is a finite Abelian group. If G has
odd order, what can you say about the 'product,' a1*a2*...*an, of all
the elements of G? What can you say about this 'product' if G has
even order? What if G is not Abelian?